1549380323-Statistical Mechanics Theory and Molecular Simulation

Non-Hamiltonian systems 47

ω ̇y=

τy

Iyy

+

(Izz−Ixx)

Iyy

ωzωx

ω ̇z=

τz

Izz

+

(Ixx−Iyy)

Izz

ωxωy. (1.11.44)

Here,ω= (0,ωx,ωy,ωz) and

S(q) =







q 1 −q 2 −q 3 −q 4

q 2 q 1 −q 4 q 3

q 3 q 4 q 1 −q 2

q 4 −q 3 q 2 q 1





. (1.11.45)

These equations of motion must be supplemented by the constraintcondition

∑

iq

2

i=

1. The equations of motion have the conserved energy

E=

1

2

[

Ixxω^2 x+Iyyωy^2 +Izzω^2 z

]

+U(q). (1.11.46)

Conservation of the energy in eqn. (1.11.46) can be shown by recognizing that the
torques can be written as

τ=−

1

2

S(q)T

∂U

∂q

. (1.11.47)

1.12 Non-Hamiltonian systems

There is a certain elegance in the symmetry between coordinates and momenta of
Hamilton’s equations of motion. Up to now, we have mostly discussed systems obeying
Hamilton’s principle, yet it is important for us to take a short detour away from this
path and discuss more general types of dynamical equations of motion that cannot
be derived from a Lagrangian or Hamiltonian function. These are referred to asnon-
Hamiltoniansystems.
Why might we be interested in non-Hamiltonian systems in the first place? To begin
with, we note that Hamilton’s equations of motion can only describe a conservative
system isolated from its surroundings and/or acted upon by an applied external field.
However, Newton’s second law is more general than this and could involve forces
that are non-conservative and, hence, cannot be derived from apotential function.
There are numerous physical systems that are characterized bynon-conservative forces,
including systems subject to frictional forces and damping effectsor the celebrated
Lorenz model that catalyzed major interest in the study of chaotic dynamics. We noted
previously that Gauss’s equations of motion (1.10.8) constitute another example of a
non-Hamiltonian system.
In order to understand how non-Hamiltonian systems may be useful in statistical
mechanics, consider a physical system in contact with a much largersystem, referred
to as abath, which regulates some macroscopic property of the physical system such
as its pressure or temperature. Were we to consider the microscopic details of the
system plus the bath together, we could, in principle, write down a Hamiltonian for